Cyclotomic Classes in a Product of Finite Abelian Groups and Applications

Cyclotomic classes of finite abelian groups have been extensively investigated for many decades, largely because of their nice algebraic structure and the breadth of their theoretical and practical applications. They naturally arise in diverse areas of mathematics, ranging from number theory and polynomial factorization to the decomposition of group algebras, and they play a particularly important role in the study of algebraic coding theory, where they provide fundamental tools for analyzing cyclic and abelian codes as well as their generalizations. Motivated by these connections, the present work concentrates on certain special types of q2-cyclotomic classes arising in a direct product of finite abelian groups, with q denoting a prime power. The main contributions of this work include a detailed characterization and enumeration of these cyclotomic classes in terms of the corresponding classes in their component groups. Several concrete examples and explicit computations are provided to illustrate the general results and to demonstrate how the cyclotomic structure behaves under group product decompositions. As an application, the paper revisits the enumeration of Hermitian complementary dual and Hermitian self-dual abelian codes defined over group algebras, presenting refined counting formulas derived from the structure of q2-cyclotomic classes. These results not only deepen our understanding of the interplay between group theory and coding theory but also contribute to the ongoing development of algebraic coding techniques grounded in group-theoretic methods.